Electrical Transformer Analysis

Mono Phase Power Transformer

©2004 Magneticasoft

A transformer is essentially a magnetic coupling device between at least two circuits (a single phase transformer) or more. The actual practice of transformer analysis and design, as shown in textbooks, is based on a very crude and arbitrary approximation of the magnetic field B (also called the induction or the flux density) inside the iron core. Further, it is assumed that B stays constant for the entirety of the core, which is a rather gross approach. By dividing the core into many cells, Magnetica is able to produce a realistic calculation of B for the totality of the core. Moreover, by introducing the concept of ferro-inductance, Magnetica generates a very accurate method for calculating of the magnetic coupling between the circuits. Consequently, it becomes possible to extend the analysis of the transformer to its behavior and performance as a function of the load.

This demonstration shows a classical electrical single phase, power stepping-down (15000/220 volts) transformer under the nominal load of 50 kilovolts ampere and 0 load (open circuit). The iron core uses a typical high permeability, grains not oriented (GNO) silicon steel. This type of material, known as electrical steel in industry, is commonly used for small power line transformers as well as for electrical motors.

The proposed symbolic names of these magnetic systems are  transfo1 and transfo2.

Magnetic System Definition

This sets the length in millimeters and the field in teslas and displays the current units.

MagneticaUserUnits[LengthUnit -> Milli Meter, 
        FieldUnit -> Tesla];

MagneticaCurrentUnits[]

This is the preliminary data for the full load transformer. The electrical current intensity is set at 3.3 amperes in the primary circuit and 222.9 amperes in the secondary circuit.

\(\(\(transfo1 = 
        AXSMagneticSystem[{AXSSolenoid[CopperOFHC, 215, 265, 348, 0, 3380, 
              3.32426, FillingFactor -> 0.6], 
            AXSSolenoid[CopperOFHC, 162, 212, 348, 0, 50, 222.901, 
              FillingFactor -> 0.6], 
            AXSSoftMagneticComponent[SiSteel2, 0, 150, 350, 0, 
              FillingFactor -> 1], 
            AXSSoftMagneticComponent[SiSteel2, 0, 267, 50, \(-201\), 
              FillingFactor -> 1], 
            AXSSoftMagneticComponent[SiSteel2, 0, 267, 50, 201, 
              FillingFactor -> 1], 
            AXSSoftMagneticComponent[SiSteel2, 267, 327, 452, 0, 
              FillingFactor -> 1]}, 
          SystemLabel -> "network transformer, iron core, effective load"];\)\(
\)
    \)\)

This is the preliminary data for the zero (quasi) load transformer. The electrical current intensity is set at 1 ampere in the primary circuit and 0.5 ampere in the secondary circuit.

transfo2 = 
        AXSMagneticSystem[{AXSSolenoid[CopperOFHC, 215, 265, 348, 0, 3380, 1, 
              FillingFactor -> 0.6], 
            AXSSolenoid[CopperOFHC, 162, 212, 348, 0, 50, 0.5, 
              FillingFactor -> 0.6], 
            AXSSoftMagneticComponent[SiSteel2, 0, 150, 350, 0, 
              FillingFactor -> 1], 
            AXSSoftMagneticComponent[SiSteel2, 0, 267, 50, -201, 
              FillingFactor -> 1], 
            AXSSoftMagneticComponent[SiSteel2, 0, 267, 50, 201, 
              FillingFactor -> 1], 
            AXSSoftMagneticComponent[SiSteel2, 267, 327, 452, 0, 
              FillingFactor -> 1]}, 
          SystemLabel -> "network transformer, iron core, initial zero \
load"];

Drawing of the Magnetic Systems

DrawSystem2D[transfo1];

Material Properties

The function Properties shows the physical and magnetic properties of the material, and the function MaterialMagnetizationCurves displays the magnetic properties as a function of the exciting magnetic field (called the magnetizing field to express its action on the matter).

Magnetization is a local property of the matter that arises from the tiny dipoles of the atomic substances. Volume magnetization, which represents the density of cooperative dipoles, has the same dimensional unit as the magnetic field. Therefore, it can be expressed in any relevant unit: gauss, telsa, ampere per meter, and so on.

The silicon steel shown has a silicon content in the middle of the electrical steel range. The relative permeability reaches the 6300 value and the saturation magnetization is 2.07 teslas.

Properties[SiSteel2]

MaterialMagnetizationCurves[SiSteel2, {M, Permeability}]

MaterialMagnetizationCurves[SiSteel2, {M, Permeability}, 
        MagnetizingFieldRange ->  {0.0001, 1}, 
        PlotScale -> LogLinearMap];

This LogLinear plot of the magnetization displays the saturation effect that takes place when the magnetizing field reaches 0.001 teslas (10 gauss) with a subsequent decrease of the permeability.

Punctual Magnetic Field Calculation

The magnetic field is calculated at the center of the magnetic structure that is inside the central iron bar. The total local magnetic field inside the material (B, called induction or traditionally called flux density) arises both from the magnetization (M) and from the local magnetizing field (Hm). The law of composition of the magnetic field is simply B = M + H (using the proper magnetic field units and considering that this is a vectorial formula). Outside the material, neglecting the air magnetization (very small), B = H (H is also called field intensity).

The function FieldPointOrigin shows the complete magnetic field contributions of the various components, primary and secondary windings, and elements of the ferromagnetic structure at the center of the central core.

FieldPointOrigin[transfo1, ComponentsList -> All]

The total flux density reaches 2.09 teslas from which 1.87 come directly from the central iron core.

FieldPointOrigin[transfo2, ComponentsList -> All]

At zero load, the flux density is reduced to 0.65 tesla of which 0.58 comes directly from the central iron core.

Iron Core Magnetization

The functional magnetization is the local value of the magnetization inside the ferromagnetic structure. The results can be expressed either in tabular form with B (flux density), M (magnetization), and H (magnetizing field) for each cell of the component or by curves as a function of cell position. With the option FullDetails -> True, the complete picture of the local magnetization is expressed for the vector elements and magnitude of the various physical parameters.

MagnetizationTable[transfo1, CellsList -> {{43, 48}}, 
      FullDetails -> True]

Analyzing some typical cells from the central core for the full load transformer, it appears that the flux density is 1.27 teslas while the magnetizing field is 0.36 millitesla (3.6 gauss or 285 amperes per meter). The relative permeability stands at 3546.

MagnetizationTable[transfo2, CellsList -> {{43, 48}}, 
      FullDetails -> True]

Analyzing the same cells in the central core for the zero load transformer, it appears that the flux density is now only 0.48 tesla while the magnetizing field is 0.086 millitesla (0.86 gauss or 6.84 amperes per meter). The relative permeability stands at 5582.

MagnetizationCurves[transfo1, {Bz}, PlotRows -> {1, 2, 3}, 
      ComponentsList -> {3}]

This graph shows the axial variation of the flux density in the central iron core at full load.

MagnetizationCurves[transfo2, {Bz}, PlotRows -> {1, 2, 3}, 
      ComponentsList -> {3}]

This graph shows the axial variation of the flux density in the central iron core at zero load.

Inductance Calculation

As stated earlier, a transformer is a device based on the magnetic coupling between two electrical circuits. The inductance is a measurement of the action of the magnetic linking flux between circuits, normalized to the current intensity. The linking flux arises from both the direct flux generated by the circuit (air core flux) and the induced flux in the ferromagnetic structures (ferro-inductance flux). Magnetica is able to calculate precisely the specific contribution of the iron core, by simple summation of the air core inductance and the total linking flux (iron core flux). (The method of calculation is patented.)

Transformer Full Load

The following verbose function VerboseInductanceMatrix provides a textual, explanatory, and formatted output for the total inductance when the transformer is fully loaded.

VerboseInductanceMatrix[transfo1, InductanceUnit -> Milli Henry, 
      ContributionType -> Total]

InductanceMatrix[transfo1, InductanceUnit -> Milli Henry, 
          ContributionType -> Proper][[1]] // 
      MatrixForm

This data corresponds to the air core contribution.

InductanceMatrix[transfo1, InductanceUnit -> Milli Henry, 
          ContributionType -> Ferro][[1]] // MatrixForm

This data corresponds to the contribution of the ferromagnetic core. It is preponderant both in terms of self-inductance and mutual inductance.

Transformer Zero Load

The same calculations are applied to the transformer at zero load.

VerboseInductanceMatrix[transfo2, InductanceUnit -> Milli Henry, 
      ContributionType -> Total]

InductanceMatrix[transfo2, InductanceUnit -> Milli Henry, 
          ContributionType -> Proper][[1]] // 
      MatrixForm

This data corresponds to the air core contribution. It is strictly identical to the data for the full load.

InductanceMatrix[transfo2, InductanceUnit -> Milli Henry, 
          ContributionType -> Ferro][[1]] // MatrixForm

This data corresponds to the contribution of the ferromagnetic core. It appears even more preponderant than for the case of a full load.

Transformer Analysis

TransformerAnalysis is a specific Magnetica function devoted to the analysis of a single phase transformer. It applies to magnetic systems such as those defined earlier. However, additional complementary data is necessary for this analysis. It is assumed that a perfect sinusoidal voltage at a constant frequency and amplitude is applied to the primary circuit (the default frequency is 50 hertz). The load is given for the secondary circuit assuming this is an active load. The secondary voltage is a nominal value, provided only for informational purposes. Then the application of the function generates a detailed analysis of the transformer for the specified load.

Transformer Full Load

The verbose function is used to obtain directly a textual, explanatory, and formatted output. The load is taken at 200 amperes.

VerboseTransformerAnalysis[transfo1, {1, 15000}, {2, 220}, 
      200]

The main results are a secondary calculated voltage of 219.2 volts versus 220 volts nominal value, a power factor (cosφ) of 0.90, and 98% power efficiency.

The next case assumes the same load but with a 60-hertz primary voltage.

VerboseTransformerAnalysis[transfo1, {1, 15000}, {2, 220}, 200, 
      InputFrequency -> 60]

Under a frequency of 60 hertz, the main results are a secondary calculated voltage of 219.2 volts versus 220 volts nominal value, a power factor (cosφ) of 0.92, and 97.9% power efficiency.

Transformer Zero Load

This case assumes a 0.5 ampere secondary active load (at 50 hertz).

VerboseTransformerAnalysis[transfo2, {1, 15000}, {2, 220}, 
      0.5]

At 50 hertz, the main results are a secondary calculated voltage of 220.5 volts versus 220 volts nominal value, a power factor (cosφ) of 0.014, and total losses of 113 watts.

The next case assumes that the secondary circuit is open (0 ampere).

VerboseTransformerAnalysis[transfo2, {1, 15000}, {2, 220}, 
      0]

At 50 hertz, the main results are a secondary calculated voltage of 220.5 volts versus 220 volts nominal value, a power factor (cosφ) of 0.005, and total losses of 87 watts.